Construction Of Reproducing Kernel Function In Two Hilbert Spaces And Numerical Approximation

Reproducing kernel theory has been widely applied in general metric geometry, numerical analysis, topological group, partial differential equation theory, fluid mechanics, probability mathematical statistics and other fields up to the present. The reproducing kernel space is more ideal space frame of the research numerical analysis, the most basic fetch value operation has a continuous express in the numerical analysis. Only because that the discrete numerical problems can be expressed by a continuous property, make the optimization of all kinds of numerical problems is possible. Because reproducing kernel space have many excellent properties, and reproducing kernel function plays a vital role for studying the property of the reproducing kernel space. Since this article focuses on how to constructing reproducing kernel function using the special skill of the reproducing kernel theory.In Hⁿ[ a , b ]space, firstly, the reproducing kernel function is constructed with Green function and the fundamental theorems of linear transforms in subspace of Hⁿ[ a , b ] space. Secondly, the reproducing kernel function is given using the sum operation of the reproducing kernel function in Hⁿ[ a , b ]space. Finally, the explicit expression of the reproducing kernel function are given in H 2[0,1]space. In H₀ⁿ [ a , b ]space, firstly, the reproducing kernel function is constructed with Green function and the definition of the reproducing kernel function in H₀ⁿ [ a , b ] space. Secondly, the interpolation spline function of H₀ⁿ [ a , b ] space is given using the reproducing kernel function of H₀ⁿ [ a , b ] space, the interpolation spline function as the operator is to the projection operator of its finite dimensional invariant subspace. Then the expression fator of the projection operator is given, and the expression fator of the projection operator is reproducing kernel function in subspace of H₀ⁿ [ a , b ]space. Finally, the best approximation of linear functional of H₀ⁿ [ a , b ] space is given with the interpolation spline function of H₀ⁿ [ a , b ] space, the convergence of linear functional of H₀ⁿ [ a , b ] space is discussed. The numerical example show that the method is effective.